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Re: zeta functions for Psmooth numbers (?)
Posted:
Mar 14, 2011 2:56 PM


David Bernier wrote: > Suppose P is some infinite subset of the primes {2, 3, 5, 7, ... }. > A positive integer n is called Psmooth if and only if all its prime > factors belong to P. > > For s in C with Re(s)> 1, one defines a "zeta_P" function as follows: > > zeta_P (s) := sum_{n >= 1 and n a Psmooth number} n^(s) . > > I'm wondering if one has a valid Eulerproduct such as: > > > sum_{n >= 1 and n a Psmooth number} n^(s) = prod_{p in P} 1/(1p^(s)) ? > > Would there be a zeta_P functional equation? > > Would there be a functional equation? > > For a specific example of a P with a very sparse complement relative to > the primes, we might consider the very sparse infinite set of primes > > T defined by: > > 1 Let c_1 be the least prime greater than Graham's number G with G as > referenced in the Wikipedia article on Graham's number: > < http://en.wikipedia.org/wiki/Graham's_number > > refers to an article on Ramsey theory by Graham & Rothschild (1971) > and a number " G = f^64 (4)" which was described in Martin Gardner's > Mathematical Games column for November 1977. > Wikipedia: > << This weaker upper bound [i.e. G as above] , attributed to some > unpublished work of Graham, was eventually published (and dubbed > Graham's number) by Martin Gardner, in [Scientific American, > "Mathematical Games", November 1977]. >> > > > 2 If c_k has been defined, k >=1, let > c_{k+1} := the least prime greater than 2^(c_k) . > > Then the sparse infinite set of primes T is defined by > T = { natural numbers n such that n=c_k for some k in N^* . }. > > So T is a very sparse infinite set of primes. Now we can let > P = Primes \ T , '\' denoting settheoretical difference, and > Primes being the set of positive integers greater than 1 > with exactly two (positive) divisors. > > Discussion numbers that aren't Psmooth include c_1, (c_1)^2, (c_1)*(c_2) , > and so on. > > I recall below the definition of zeta_P : > > ==> zeta_P (s) := sum_{n >= 1 and n a Psmooth number} n^(s) . [ Re(s) > > 1] > > Following Harold Edwards book "Riemann's Zeta Function", > how much of the plan of (roughly) Chapters 1 through 7 could > be carried out? > >  > > Motivation: this question of a form of zeta function for Psmooth numbers > that I call "zetaP" interests me because the infinite set > P can be made toorder; for example, one could have a subset > of the Primes P such that the "primes in P"counting function > say pi_P (.) would satisfy Schoenfeld's explicit > version of von Koch's Theorem as long as x < G, > G being Graham's number: > http://en.wikipedia.org/wiki/Riemann_hypothesis#Distribution_of_prime_numbers > > > for x > G, one could have for arbitrarily large x: >  pi_P (x)  Li(x) > x^(51/100) > while still respecting  pi_P (x)  Li(x) = O(x^(51/100+epsilon)), > for any given epsilon > 0. > NB.: pi_P (x) is a count of the primes p belonging to P and less than or > equal to x . [...]
One of the simplest cases, in the same spirit, is to relax the condition on the cardinality of Primes \ P, by allowing Primes \ P to have a finite cardinality. One might as well first try to remove just one prime; if one is to remove just one prime, one can try removing the smallest, 2.
Then P = Primes \ {2}. The Psmooth integers are the odd numbers {1, 3, 5, 7, ... }.
zeta_P (s) := 1 + 1/3^s + 1/5^s + 1/7^s + 1/9^s + 1/11^s + ... , Re(s) > 1.
(a) Does zeta_P (s) have an analytic continuation? (b) What if one tries to perform EulerMaclaurin summation on zeta_P(1/2 + it) "=" 1 + 1/[3^(1/2 + it)] + 1/[5^(1/2 + it)] + 1/[7^(1/2 + it)]+ 1/[9^(1/2 + it)] + ... for t > 0 ? analytic? convergent? where are the zeros? Does one have an analog for von Mangoldt's explicit formula for the summatory Lambda function, psi(x) in Edwards, section 3.2, page 54 ? etc.
David Bernier



